[pheelicks]

The base platonic solid that Healpix is based on is the octahedron (https://en.wikipedia.org/wiki/Octahedron), which A5 uses the dodecahedron(https://en.wikipedia.org/wiki/Regular_dodecahedron).

The octahedron has a much higher angular defect (https://en.wikipedia.org/wiki/Angular_defect) than the dodecahedron, and thus when it is projected onto the sphere the cells are warped a lot. So while their areas may be the same, the shapes vary.

This article explains the geometric construction, and how it leads to the cells being a similar size and shape: https://a5geo.org/docs/technical/platonic-solids

Also from a data visualization point of view, the rectangular cells of Healpix (like S2) are arguably less pleasing to look at than hexagons/pentagons: https://h3geo.org/docs/comparisons/s2#visualization

[pama]

Not sure I understand—healpix starts from the rhombic dodecahedron and then bisects the generalizations of the 12 squares each time. Where do octahedra come into play?

[pheelicks]

My mistake, you are correct. The base solid is indeed the rhombic dodecahedron. I believe the point about the angular defect is still valid though.